There is a strong analogy between continuous sections of a topological
fibration, rational points of a variety defined over a non-closed field,
and fixed points of the action of a finite group. I will explain this
analogy as well as a number of recent results for rational points and
fixed points suggested by this analogy: a theorem of de Jong-He-Starr,
Yi Zhu's theorem, the Tian-Zong theorem, and work-in-progress of
Chenyang Xu and myself for rational points of varieties over global
function fields.